37
votes
2answers
1k views

$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was let to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free. ...
2
votes
0answers
97 views

The transfer map $H_*(BSO(3))\rightarrow H_*(BO(2))$: reference request

All cohomology and homology will be $Z/2$ coefficient. The restriction map $H^*(BSO(3))\rightarrow H^*(BO(2))$ is well-known to be the inclusion of the Dickson invariant $Z/2[w_2,w_3]$ into the ...
3
votes
1answer
216 views

Cohomology of Projective Classical Lie Groups

Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes ...
4
votes
1answer
339 views

Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups

I've recently read the following line in an interesting paper: It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...
5
votes
3answers
594 views

Continuous cohomology of semi-simple Lie group.

Let $G$ be a real connected semi-simple Lie group. Let $M$ be a finite dimensional representation of it. Are there general criteria when the continuous cohomology groups $H_{cont}^q(G,M)$ vanish? A ...
11
votes
2answers
738 views

Torsion for Lie algebras and Lie groups

This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for ...
2
votes
1answer
405 views

Generator of Lie Group cohomology in degree 3

This is my first question. Take a simple, connected, compact, simply connected Lie group $ G$ (dim $G\geq 3$). The cohomology of $G$ with integer coefficients is $H^{1,2}(G,\mathbb{Z})\cong 0$, ...
4
votes
1answer
530 views

smooth cohomology of Lie groups

Let $G$ be a Lie group and $A$ a smooth $G$-module. Define $C^n(G,A)=\{ f: G^n \to A|~f~\text{is smooth}\}$ and $\partial^n: C^n \to C^{n+1}$ by the standard formula as used in the cohomology of ...
0
votes
1answer
413 views

Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology

The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about. ...
3
votes
1answer
398 views

Relation of Lie Groups and Cohmology Theories via Formal Group Laws

There is a standard process (for example explained here) to obtain a formal group law form a complex oriented cohomology theory. For a Lie group G one can choose coordinates at the unit and expand ...
11
votes
0answers
390 views

To what extent does (co)homology of groups made discrete depend on set theory?

There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. ...
3
votes
2answers
347 views

What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups?

As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into ...
6
votes
2answers
380 views

Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?

Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...
11
votes
4answers
973 views

How to compute the (co)homology of orbit spaces (when the action is not free)?

Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...